A Pseudodynamic Approach of Seismic Active Pressure on Retaining Walls Based on a Curved Rupture Surface

Reasonable determination of the magnitude and distribution of dynamic earth pressure is one of the major challenges in the seismic design of retaining walls. Based on the principles of pseudodynamic method, the present study assumed that the critical rupture surface of backfill soil was a composite curved surface which was in combination with a logarithmic spiral and straight line.)e equations for the calculation of seismic total active thrusts on retaining walls were derived using limit equilibrium theory, and earth pressure distribution was obtained by differentiating total active thrusts.)e effects of initial phase, amplification factor, and soil friction angle on the distribution of seismic active earth pressure have also been discussed. Compared to pseudostatic and pseudodynamic methods for the determination of planar failure surface forms, the proposed method receives a bit lower value of seismic active earth pressures.


Introduction
One of the major tasks in seismic designs is the determination of dynamic earth pressures on retaining walls during an earthquake which makes the development of a realistic seismic earth pressure theory essential. Based on static Coulomb earth pressure theory, Okabe [1] and Mononobe and Matsuo [2] conducted in-depth investigations on different forms of retaining wall failure modes during earthquakes and proposed equations for the calculation of seismic earth pressures on retaining walls, known as Mononobe-Okabe (M-O) or pseudostatic method. is method considers seismic load as a simple inertial force to take into account the dynamic nature of earthquake loadings in a very approximate way. Steedman and Zeng [3] proposed a pseudodynamic method to consider certain dynamic response characteristics in a relatively simple way. eir proposed method solved the problem of seismic earth pressures on retaining walls by taking into account the phase differences and amplification effects of seismic waves by assuming the critical fracture surface of the backfill as being planar. Zeng and Steedman [4] compared pseudodynamic analysis results with those obtained experimentally from a centrifuge model to verify the accuracy of the proposed theoretical method. In the pseudodynamic method proposed by Steedman and Zeng [3], simply shear wave velocity (v s ) and horizontal seismic acceleration (k h g, where g is the acceleration due to gravity) were considered. Choudhury and Nimbalkar [5] considered primary wave velocity (v p ) and vertical seismic acceleration (k v g) based on the pseudodynamic method developed by Steedman and Zeng [3] to study the seismic earth pressures on retaining walls. Many other researchers have conducted detailed studies using the pseudodynamic method, assuming the failure surface of the backfill to be planar [6][7][8][9][10].
All the abovementioned methods assume the failure surface of backfill to be planar. Terzaghi et al. [11] found that the seismic earth pressure coefficients obtained by assuming flat rupture surfaces had large deviations from real values with calculation errors of up to 3 times higher than the real value. By assuming curved rupture surfaces, Kumar [12] deduced theoretical equations for the calculation of seismic earth pressures on the back of inclined retaining walls based on the pseudostatic method. It was found that the calculated seismic earth pressures were more reasonable when the critical rupture surface was assumed to be curved. Based on the pseudostatic method, Soubra [13], Choudhury et al. [14], and Subba Rao and Choudhury [15] assumed critical rupture surfaces as logarithmic spiral or a combination of logarithmic spiral and straight lines and derived theoretical equations for seismic earth pressures at different dip angles of retaining walls. Also, Basha and Babu [16][17][18][19] studied seismic structures using the pseudodynamic method by assuming curved critical rupture surface for backfill but did not obtain the distributions of seismic earth pressures along the depth of retaining walls. Xu et al. [20] applied the LSR (log-spiral-Rankine) model to assess active and passive seismic earth pressures and introduced local and global iteration schemes to solve the resulting highly coupled multivariate nonlinear system of equations, which was more complicated. Santhoshkumar et al. [21] computed the seismic active resistance of a slanted cantilever retaining wall holding a cohesionless backfill and did not assume a preordained failure mechanism.
To the best of our knowledge, seismic active earth pressure theory based on the pseudodynamic method by assuming curved rupture surfaces has not yet been developed. erefore, in this study, we assumed composite curve critical rupture surface for backfill and investigated the effects of the phase difference and amplification of seismic waves, soil friction angle, soil-wall friction angle, horizontal seismic acceleration coefficient, and vertical seismic acceleration coefficient. According to limit equilibrium theory, we also deduced equations for the calculation of total seismic active thrusts on the walls and obtained the distributions of active earth pressures along the depth of retaining walls. e effects of the initial phase, soil amplification factor, and soil friction angle on the distribution of seismic active earth pressure had also been discussed. e obtained results were compared with previous studies where the distributions of seismic active earth pressure were obtained by pseudostatic and pseudodynamic methods under the assumption of planar rupture surfaces.

Basic Assumptions.
Based on the principles of pseudodynamic method, in the present study, we have taken the following assumptions: (a) e back of the retaining wall was vertical and the surface of backfill was horizontal (b) e critical rupture surface of backfill was a combination of logarithmic spiral and straight line, and the rupture surface of the soil wedge passed through the heel of the retaining wall (c) e shear modulus G of backfill soil was constant along the wall (d) Backfill was homogeneous, isotropic, dry, and noncohesive soil (e) Zero-stress-boundary condition was neglected Figure 1 shows the schematic diagram of a retaining wall calculation model based on the proposed method. As mentioned above, the back of the retaining wall was assumed to be vertical with height H and backfill soil was considered to be noncohesive and the effects of amplification factor, soil friction angle, soil-wall friction angle, and horizontal and vertical seismic acceleration coefficients were taken into account. Under the action of horizontal and vertical seismic inertial forces, the horizontal and vertical seismic accelerations at any depth and any time were as follows: (1) e purpose of this study was to derive a series of equations to simplify the calculation of the distribution of seismic active earth pressure, similar to those in Rankine's earth pressure theory, M-O method, and Choudhury and Nimbalkar method [5]. Before this work, there was no report on equations for deriving seismic active earth pressure based on a curved rupture surface. erefore, we only assumed one failure mechanism to simplify the calculation of seismic active earth pressure and fixed the focus of log-spiral curve at point "O" under active seismic conditions.

Deducing Process.
e force of soil wedge OCB was first analyzed to obtain contact force N 2 between the wedges OCB and OAC (as shown in Figure 1). e mass of a thin element (dz) of the wedge OCB at depth z and the total weight of the soil wedge (shown in Figure 1 Horizontal and vertical seismic inertial forces of wedge OCB were calculated as Contact force N 2 (t) could be obtained by calculating forces on wedge OCB and considering the equilibrium of forces as Similarly, taking wedge OAC as the mechanics case study (shown in Figure 1(c)), the masses of the thin elements of wedges ODC and DAC could be written as e total weights of wedges ODC and DAC could be expressed as Horizontal and vertical seismic inertial forces acting on wedges ODC and DAC were Total seismic active thrust on the retaining wall could be obtained by resolving forces acting horizontally and vertically on wedge OAC as follows.
Considering horizontal equilibrium condition (ΣH � 0) for wedge OAC, Considering vertical equilibrium condition (ΣV � 0) for wedge OAC, where where F h and F v can be estimated as Solving equations (8) and (9), the total active thrust under seismic conditions could be obtained as Seismic active earth pressure coefficient K ae (t) was defined as

Mathematical Problems in Engineering
To obtain the value of seismic active earth pressure coefficient, the height of wedge OCB, h, should be substituted by that of the retaining wall, H. e logarithmic spiral equation was used as follows: From equation (14), we can relate h to H as follows: According to Rankine's theory, for active case, the angle made by rupture surface and horizontal line could be obtained from α � 45°+ (φ/2). e maximum value of K ae could be obtained by optimizing K ae with respect to the initial phase of t/T (0 < t/T < 1). e distribution of seismic active earth pressure was obtained by differentiating total active thrust P ae (t) as where To give physical significance to the term "zN 2 (t)/zz" in the whole depth range of the retaining wall, the upper limit of integral was changed by the mathematical calculus method and the distribution of N 2 (t) within the depth range of h was transformed into distribution within the depth range H of the retaining wall (as shown in Figure 2) as Finally, we got: Because the polar equation of logarithmic spiral could not be directly related to the rectangular coordinate equation, for the length of a thin element on wedge DAC, the solution given in this study used 1, x, x 2 , x 3 (Table 1) could be expressed as

Analysis and Comparison of Effective Factors
In this study, different parameters such as H/λ s � 0.5 and H/λ p � 0.03 have been used in the study of Maskar et al. [9], and these values do not represent actual values. e readers can substitute exact values according to actual situation. We have focused on the process and method of determining the distribution of active earth pressures on retaining walls under seismic conditions with curved failure surfaces. In the following simple analyses, the distributions and variations of earth pressure were determined considering the effects of initial phase (t/T) change of seismic acceleration in one wave period, amplification factor f a , and soil friction angle φ. e effects of other seismic and backfill soil parameters on the distribution of soil pressure have not been discussed here. All calculations in this paper were carried out by MATLAB software [21]. Figure 3 shows the distribution and variation of earth pressure along the depth z of retaining wall with initial phase (t/T) change in seismic acceleration in one wave period. Different initial phase t/T values of 0, 0.25, 0.50, and 0.75 were applied here, including when K ae reached the maximum value (t/T � 0.60). As can be seen in Figure 3, the value of total active earth thrust in one period P ae (t) was first increased and then decreased with the increase of t/T (based on the area enclosed by the earth pressure distribution curve and coordinate axis). K ae had its minimum value when t/T � 0.10, and its maximum value was obtained when t/T � 0.60. At the bottom of the retaining wall (z/H � 1), earth pressure was first increased and then decreased with time, reaching its maximum value when t/T � 0.50. Earth pressure was approximately 59% of its maximum value when t/T � 0. Since dynamic pressure was increased nonlinearly with depth, the position of dynamic thrust varied with time ( Figure 3). When the earthquake occurs, the distribution of earth pressure changes with time (or phase) similar to the phenomenon is observed in this part. We took the time when the total active thrust reached its maximum value due to the critical seismic state of the retaining wall.

Effect of Amplification Factor (f a ).
To facilitate the analysis of the distribution of seismic active earth pressure with earthquake or soil parameters, the distribution of the earth pressure was the distribution when K ae reaches its maximum value. Figure 4 shows the distribution of seismic active earth pressure along the depth of the retaining wall for different values of the amplification factor. As shown in Figure 4, by changing the value of the amplification factor, the shape of the earth pressure distribution curve was slightly changed. By increasing the value of f a , total active earth thrust was also gradually increased. It was found that the maximum value of K ae (K ae (t)max) when f a � 1 was approximately 84% of its corresponding value when f a � 2.
At the bottom of retaining wall (z/H � 1), the maximum value of earth pressure (p ae (t)max) when f a � 1 was approximately 90% of its corresponding value when f a � 2. is indicated that the vibration amplification effect of backfill had a relatively slight impact on seismic active earth pressure on the retaining wall. erefore, to ensure the safety of the structure, vibration amplification effect of soil had to be taken into account while designing safe seismic structures. Figure 5 shows the distribution of soil pressure along the depth of the retaining wall for different friction angles φ of backfill soil. It can be clearly seen in the figure that the variation of friction angle φ had a great effect on seismic active earth pressure on the retaining wall. e value of K ae (t)max when φ � 50°was approximately 38% of its corresponding value when φ � 20°. At the bottom of the retaining wall (z/H � 1), the maximum value of earth pressure (p ae (t)max) when φ � 50°was about 32% of its corresponding value when φ � 20°. erefore, it was concluded that selection of proper values for soil friction angle φ was very important in seismic safe design since improper values could cause great calculation errors and ultimately affect the safety of retaining structures under seismic forces. Figure 6 compares our results with those obtained from M-O (Mononobe-Okabe) and C-N (Choudhury-Nimbalkar) methods for the distribution of active earth pressure on a retaining wall under seismic conditions when f a � 1, k h � 0.2, k v � 0.5k h , ϕ � 30°, δ � 0.5ϕ, H/λ s � 0.5, and H/λ p � 0.03. Like C-N method, the earth pressure distribution curve obtained in the study presented nonlinear characteristics. In general, based on pseudodynamic method, the maximum value of total active earth thrust calculated in this study for curved critical rupture surface was lower than that obtained from the C-N It can be seen in Figure 6 that, under the same